Sharp Sobolev inequality involving a critical nonlinearity on a boundary
Keywords
Neumann problem, critical Sobolev exponent, topological linkingAbstract
We consider the solvability of the Neumann problem for the equation $$ -\Delta u+\lambda u =0, \quad \frac {\partial u}{\partial \nu}=Q(x)|u|^{q-2}u $$ on $\partial \Omega$, where $Q$ is a positive and continuous coefficient on $\partial \Omega$, $\lambda$ is a parameter and $q= {2(N-1)}/{(N-2)}$ is a critical Sobolev exponent for the trace embedding of $H^1(\Omega)$ into $L^q(\partial \Omega)$. We investigate the joint effect of the mean curvature of $\partial \Omega$ and the shape of the graph of $Q$ on the existence of solutions. As a by product we establish a sharp Sobolev inequality for the trace embedding. In Section 6 we establish the existence of solutions when a parameter $\lambda$ interferes with the spectrum of $-\Delta$ with the Neumann boundary conditions. We apply a min-max principle based on the topological linking.Downloads
Published
2005-03-01
How to Cite
1.
CHABROWSKI, Jan and YANG, Jianfu. Sharp Sobolev inequality involving a critical nonlinearity on a boundary. Topological Methods in Nonlinear Analysis. Online. 1 March 2005. Vol. 25, no. 1, pp. 135 - 153. [Accessed 5 November 2024].
Issue
Section
Articles
Stats
Number of views and downloads: 0
Number of citations: 0